how to move face along an axis without affecting other faces?
Each edge is defined by 2 vertices, and each face is defined by at least 3 vertices. On the screenshot below, moving selected face is moving 4 vertices it is defined by. Since this face shares 1 vertex with the edge marked as (1) and 1 vertex with the edge marked as (2), moving this face, will warp those edges, because you will move half of the vertices defined by them:

You can see which edges a vertex is a part of, by selecting it, and looking for gradient lines:

If, while having the same face selected as on pic. 1, you right-click and choose Split, then the vertices of the selection will be duplicated, without any connections to vertices outside of the selection. Selecting a vertex again should make the difference obvious:

Selecting all duplicates (or just everything), pressing M, B, will merge the vertices, by removing all but one vertex at each spot and moving all links of those vertices to that one.
However, you probably never used Split. What you did use was Extrude, which takes a face A and duplicates it to face B, but also (unlike Split) for each vertex of face B it creates a link to the original vertex it duplicated in face A (as well as it creates faces between edges created this way). Then the new face is being moved along its normal. And what you probably did was Extrude, but then right-click, leaving the new face in the same spot as the old one, but no longer directly connected to edges (1) and (2) on the first picture. This is why you could move that face without warping those edges, and this is why so many people advised to simply extrude the other face - if you want to keep the consistency with the other face, but it is not the correct way to build &quot;proper&quot; topology.
Moving the original face:

Extruding the original face:

Extruding the original face, but right-clicking to leave it at original coordinates; deselecting it, selecting it and then moving it: